Calculus Examples

$\frac{d y}{d x} + 4 y = y (e^{x} + 4)$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify $y (e^{x} + 4)$ .

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Step 1.1.1.1

Rewrite.

$\frac{d y}{d x} + 4 y = 0 + 0 + y (e^{x} + 4)$

Step 1.1.1.2

Simplify by adding zeros.

$\frac{d y}{d x} + 4 y = y (e^{x} + 4)$

Step 1.1.1.3

Apply the distributive property.

$\frac{d y}{d x} + 4 y = y e^{x} + y \cdot 4$

Step 1.1.1.4

Move $4$ to the left of $y$ .

$\frac{d y}{d x} + 4 y = y e^{x} + 4 y$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Subtract $4 y$ from both sides of the equation.

$\frac{d y}{d x} = y e^{x} + 4 y - 4 y$

Step 1.1.2.2

Combine the opposite terms in $y e^{x} + 4 y - 4 y$ .

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Step 1.1.2.2.1

Subtract $4 y$ from $4 y$ .

$\frac{d y}{d x} = y e^{x} + 0$

Step 1.1.2.2.2

Add $y e^{x}$ and $0$ .

$\frac{d y}{d x} = y e^{x}$

Step 1.2

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y e^{x})$

Step 1.3

Cancel the common factor of $y$ .

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Step 1.3.1

Factor $y$ out of $y e^{x}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y (e^{x}))$

Step 1.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y e^{x})$

Step 1.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = e^{x}$

Step 1.4

Rewrite the equation.

$\frac{1}{y} d y = e^{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int e^{x} d x$

Step 2.2

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int e^{x} d x$

Step 2.3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$\ln (| y |) + C_{1} = e^{x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (| y |) = e^{x} + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{e^{x} + C}$

Step 3.2

Rewrite $\ln (| y |) = e^{x} + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{e^{x} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{e^{x} + C}$ .

$| y | = e^{e^{x} + C}$

Step 3.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{e^{x} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{e^{x} + C}$ as $e^{e^{x}} e^{C}$ .

$y = \pm e^{e^{x}} e^{C}$

Step 4.2

Reorder $e^{e^{x}}$ and $e^{C}$ .

$y = \pm e^{C} e^{e^{x}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{e^{x}}$